Semiclassical analysis of the nuclear-magnetic-resonance absorption of a tetrahedrally coordinated four-spin-1/2 system

A semiclassical theory of proton magnetic-resonance absorption of the N${\mathrm{H}}_4^{+}$ ion in solids is presented. In this theory, the lattice variables are treated as classical functions of time, while the nuclear spins are treated quantum mechanically. No effects of the exclusion principle are considered. Theoretical spectra are calculated in the limit of fast motion, when the N${\mathrm{H}}_4^{+}$ ion is either rotating uniformly or undergoing random reorientations about a single, fixed symmetry axis. If the ion is assumed to reorient rapidly about a single $C_3$ axis, a satisfactory agreement between theoretical and experimental spectra is obtained for N${\mathrm{H}}_4$V${\mathrm{O}}_3$ at 77 K. The theoretical line shape also agrees quite well with the experimental line shape of N${\mathrm{H}}_4$I at 4 K, if the N${\mathrm{H}}_4^{+}$ ion is considered to be rotating around a fixed $C_4$ axis with a single frequency equal to 1.5 times the N${\mathrm{H}}_4$ dipolar frequency. It is shown that this "classical" frequency, which causes satellites to appear in the absorption spectrum, is a measure of the torsional ground-state splitting of the N${\mathrm{H}}_4^{+}$ ion in the crystal field.